A closely related question is to quantify the precise range of
exponents \(\alpha \in {\mathbb {R}}\)  for which
the weight \(w_{\alpha ,E}(x)=\operatorname{dist}(x,E)^{-\alpha }\) 
belongs to the Muckenhoupt class \(A_p\)  for a given \(1\le p<\infty \) .
If \(E\subset {\mathbb {R}}^n\)  is porous, it
follows from [1]}
that
\(w_{\alpha ,E}\in A_1\)  if and only if \(0\le \alpha <n-\operatorname{dim_A}(E)\) , and
\(w_{\alpha ,E}\in A_p\) , for \(1<p<\infty \) ,
if and only if
\((1-p)(n-\operatorname{dim_A}(E))<\alpha <n-\operatorname{dim_A}(E).\) 
